It should be noted that problem (REF ) and (REF )
has a unique solution \(x(t)=x(t;\alpha ,\xi )\)  defined on
\([\alpha -\sigma ,\infty )\)  which is piecewise continuous with points
of discontinuity of the first kind, namely \(t_k, k\in \mathbb {Z}\) ,
at which it is left continuous and the following relations are satisfied [1]}
\(x\left(t^-_k\right)=x(t_k),\ \Delta x(t_k):=x\left(t_k^+\right)-x\left(t^-_k\right)=\gamma _kx\left(t_k^-\right)+\delta _k.\) 
